Interacting Fock spaces and Gaussianization of probability measures

We prove that any probability measure on R, with moments of all orders, is the vacuum distribution, in an appropriate interacting Fock space, of the field operator plus tin the nonsymmetric case) a function of the number operator. This follows from a canonical isomorphism between the L-2-space of the measure and the interacting Fock space in which the number vectors go into the orthogonal polynomials of the measure and the modified held operator into the multiplication operator by the x-coordinate. A corollary of this is that all the momenta of such a measure are expressible in terms of the Szego-Jacobi parameters, associated to its orthogonal polynomials, by means of diagrams involving only noncrossing pair partitions land singletons, in the nonsymmetric case). This means that, with our construction, the combinatorics of the momenta of any probability measure (with all moments) is reduced to that of a generalized Gaussian. This phenomenon we call Gaussianization. Finally we define, in terms of the Szego-Jacobi parameters, a new convolution among probability measures which we call universal because any probability measure (with all moments) is infinitely divisible with respect to this convolution. All these results will be extended to the case of many (in fact infinitely many) variables in a future paper.